Table of contents

0. Review of College Algebra4h 43m
1. Measuring Angles40m
2. Trigonometric Functions on Right Triangles2h 5m
3. Unit Circle1h 19m
4. Graphing Trigonometric Functions1h 19m
5. Inverse Trigonometric Functions and Basic Trigonometric Equations1h 41m
6. Trigonometric Identities and More Equations2h 34m
7. Non-Right Triangles1h 38m
8. Vectors2h 25m
9. Polar Equations2h 5m
10. Parametric Equations1h 6m
11. Graphing Complex Numbers1h 7m

8. Vectors

Dot Product

Trigonometry

8. Vectors

Dot Product

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1:43 minutes

Problem 40

Textbook Question

In Exercises 40–41, use the dot product to determine whether v and w are orthogonal.v = 12i - 8j, w = 2i + 3j

Verified step by step guidance

Step 1: Recall that two vectors \( \mathbf{v} \) and \( \mathbf{w} \) are orthogonal if their dot product is zero.

Step 2: Write the formula for the dot product of two vectors \( \mathbf{v} = a_1\mathbf{i} + b_1\mathbf{j} \) and \( \mathbf{w} = a_2\mathbf{i} + b_2\mathbf{j} \): \( \mathbf{v} \cdot \mathbf{w} = a_1a_2 + b_1b_2 \).

Step 3: Substitute the given vectors \( \mathbf{v} = 12\mathbf{i} - 8\mathbf{j} \) and \( \mathbf{w} = 2\mathbf{i} + 3\mathbf{j} \) into the dot product formula.

Step 4: Calculate the dot product: \( 12 \times 2 + (-8) \times 3 \).

Step 5: Determine if the result from Step 4 is zero. If it is, then \( \mathbf{v} \) and \( \mathbf{w} \) are orthogonal.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Dot Product

The dot product, also known as the scalar product, is a mathematical operation that takes two vectors and returns a single scalar value. It is calculated by multiplying the corresponding components of the vectors and summing the results. For vectors v = ai + bj and w = ci + dj, the dot product is given by v · w = ac + bd. This operation is crucial for determining the angle between vectors and checking for orthogonality.

Orthogonal Vectors

Two vectors are considered orthogonal if they are perpendicular to each other, which occurs when their dot product equals zero. This property is significant in various applications, including physics and engineering, as it indicates that the vectors do not influence each other in their respective directions. In the context of the given vectors, checking for orthogonality involves calculating their dot product and verifying if it equals zero.

Vector Components

Vectors in a two-dimensional space can be expressed in terms of their components along the x-axis and y-axis, typically represented as v = ai + bj. Here, 'a' and 'b' are the scalar components that indicate the vector's magnitude in the respective directions. Understanding vector components is essential for performing operations like the dot product, as it allows for the systematic calculation of the interaction between vectors in a Cartesian coordinate system.